Nonlinear Spectrum and Fixed Point Index for a Class of Decomposable Operators

We study a class of nonlinear operators that can be written as the composition of a linear operator and a nonlinear map. We obtain results on fixed point index based on parameters that are related to the definitions of nonlinear spectra. As a particular case, existence of positive solutions for a second-order differential equation with separated boundary conditions is proved. The result also provides a spectral interval for the corresponding Hammerstein integral operator.

Decomposability of Krein space operators

In this paper, we review some properties in the local spectral theory and various subclasses of decomposable operators. We prove that every Krein space selfadjoint operator having property (?) is decomposable, and clarify the relation between decomposability and property (?) for J-selfadjoint operators. We prove the equivalence of these properties for J-selfadjoint operators T and T* by using their local spectra and local spectral subspaces.

Approximation and Similarity Classification of Stably Finitely Strongly Irreducible Decomposable Operators

Let 𝓗 be a complex separable Hilbert space and ℒ(𝓗) denote the collection of bounded linear operators on 𝓗. In this paper, we show that for any operator A ∈ ℒ(𝓗), there exists a stably finitely (SI) decomposable operator A∈, such that ‖A−A∈‖ 𝓗 ∈ andA′(A∈)/ rad A′(A∈) is commutative, where rad A′(A∈) is the Jacobson radical of A′(A∈). Moreover, we give a similarity classification of the stably finitely decomposable operators that generalizes the result on similarity classification of Cowen–Douglas operators given by C. L. Jiang.